The relation between motion in −1/ r and r 2 potentials, known since Newton, can be demonstrated by the substitution r → r 2 in the classical/quantum radial equations of the Kepler/Hydrogen problems versus the harmonic oscillator. This suggests a duality-type relationship between these systems. However, when both radial and angular components of these systems are included the possibility of a true duality seems to be remote. Indeed, investigations that explored and generalized Newton’s radial relation, including algebraic approaches based on noncompact groups such as SO(4,2), have never exhibited a full duality consistent with Newton’s. On the other hand, 2T-physics predicts a host of dualities between pairs of a huge set of systems that includes Newton’s two systems. These dualities take the form of rather complicated canonical transformations that relate the full phase spaces of these respective systems in all directions. In thi...

中文翻译：

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氢原子与振荡器系统之间的对偶性（通过隐藏的SO（d，2）对称性和2T物理学）

自牛顿以来就知道的在-1 r和r 2势中的运动之间的关系可以通过开普勒/氢问题相对于谐波振荡器的经典/量子径向方程式中的r→r 2来证明。这暗示了这些系统之间的对偶类型关系。但是，当同时包含这些系统的径向和角向分量时，实现真正对偶的可能性似乎很小。确实，探索和推广牛顿径向关系的研究，包括基于非紧凑群（如SO（4,2））的代数方法，从未表现出与牛顿一致的完全对偶性。另一方面，2T物理学预测包括牛顿的两个系统在内的庞大系统对之间的大量对偶。这些对偶采取相当复杂的规范变换的形式，这些规范变换将这些各个系统的完整相空间在各个方向上关联起来。在...